The recent discovery of habitable exoplanets around Proxima Centauri and TRAPPIST-1 has attracted much attention due to their potential for hosting life. We delineate a simple model that accurately describes the evolution of biological diversity on Earth. Combining this model with constraints on atmospheric erosion and the maximal evolutionary timescale arising from the star's lifetime, we arrive at two striking conclusions: (i) Earth-analogs orbiting low-mass M-dwarfs are unlikely to be inhabited, and (ii) K-dwarfs and some G-type stars are potentially capable of hosting more complex biospheres than the Earth. Hence, future searches for biosignatures should prioritize planets around K-dwarf stars.